arXiv subjects

Lorenz Panny

Publications and source records attributed to Lorenz Panny.

4 recordsLinked to original sources

arXiv2026

The SEA algorithm for endomorphisms of supersingular elliptic curves

For a prime $p{\,>\,}3$ and a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$ with ${j(E)\notin\{0,1728\}}$, consider an endomorphism $α$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We prove that the trace of $α$ may be computed in $O(n^4(\log n)^2 + dLn^3)$ bit operations, where $n{\,=\,}\log(p)$, using a generalization of the SEA algorithm for computing the trace of the Frobenius endomorphism of an ordinary elliptic curve. When $L\in O(\log p)$ and $d\in O(1)$, this complexity matches the heuristic complexity of the SEA algorithm. Our theorem is unconditional, unlike the complexity analysis of the SEA algorithm, since the kernel of an arbitrary isogeny of a supersingular elliptic curve is defined over an extension of constant degree, independent of $p$. We also provide practical speedups, including a fast algorithm to compute the trace of $α$ modulo $p$.

math.NT↗

arXiv2025

Evaluation of Modular Polynomials from Supersingular Elliptic Curves

We present several new algorithms to evaluate modular polynomials of level $\ell$ modulo a prime $p$ on an input $j$. More precisely, we introduce two new generic algorithms, sharing the following similarities: they are based on a CRT approach; they make use of supersingular curves and the Deuring correspondence; and, their memory requirements are optimal. The first algorithm combines the ideas behind a hybrid algorithm of Sutherland in 2013 with a recent algorithm to compute modular polynomials using supersingular curves introduced in 2023 by Leroux. The complexity (holding around several plausible heuristic assumptions) of the resulting algorithm matches the $\Tilde{O}(\ell^3 \log^{3} \ell + \ell \log p)$ time complexity of the best known algorithm by Sutherland, but has an optimal memory requirement. Our second algorithm is based on a sub-algorithm that can evaluate modular polynomials efficiently on supersingular $j$-invariants defined over $\Fp$, and achieves heuristic complexity quadratic in both $\ell$ and $\log j$, and linear in $\log p$. In particular, it is the first generic algorithm with optimal memory requirement to obtain a quadratic complexity in~$\ell$. Additionally, we show how to adapt our method to the computation of other types of modular polynomials such as the one stemming from Weber's function. Finally, we provide an optimised implementation of the two algorithms detailed in this paper, though we emphasise that various modules in our codebase may find applications outside their use in this paper.

math.NT↗

arXiv2021

Improved torsion point attacks on SIDH variants

SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a secret isogeny to a subgroup of the curve (torsion point information). Petit (2017) was the first to demonstrate that torsion point information could noticeably lower the difficulty of finding secret isogenies. In particular, Petit showed that "overstretched" parameterizations of SIDH could be broken in polynomial time. However, this did not impact the security of any cryptosystems proposed in the literature. The contribution of this paper is twofold: First, we strengthen the techniques of Petit by exploiting additional information coming from a dual and a Frobenius isogeny. This extends the impact of torsion point attacks considerably. In particular, our techniques yield a classical attack that completely breaks the n-party group key exchange of Azarderakhsh et al. for 6 parties or more, and a quantum attack for 3 parties or more that improves on the best known asymptotic complexity. We also provide a Magma implementation of our attack for 6 parties. We give the full range of parameters for which our attacks apply. Second, we construct SIDH variants designed to be weak against our attacks; this includes backdoor choices of starting curve, as well as backdoor choices of base-field prime. We stress that our results do not degrade the security of, or reveal any weakness in, the NIST submission SIKE.

math.NT↗

arXiv2021

Quantum Equivalence of the DLP and CDHP for Group Actions

In this short note we give a polynomial-time quantum reduction from the vectorization problem (DLP) to the parallelization problem (CDHP) for group actions. Combined with the trivial reduction from par-allelization to vectorization, we thus prove the quantum equivalence of both problems.

cs.CR↗